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arXiv · 2503.14073

Enumerating k-connected blocks and gk-connected blocks in words

Walaa Asakly·Noor Kezil
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Abstract

We define two new statistics on words: the k-connector and the gk-connector. For a word $π= π_1π_2\cdotsπ_n$ of length $n$ over the alphabet $[k]$, a k-connector is defined as an ordered pair $(π_j, π_{j+1})$ where $1 \leq j \leq n-1$ and $π_j + π_{j+1} = k$. Conversely, a gk-connector is defined as an ordered pair $(π_j, π_{j+1})$ where $1 \leq j \leq n-1$ and $π_j + π_{j+1} > k$. We investigate the enumeration of partitions based on these statistics, providing generating functions and explicit formulas.

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Walaa Asakly, Noor Kezil. 2025-03-18. Enumerating k-connected blocks and gk-connected blocks in words. https://arxiv.org/abs/2503.14073

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